1. **Stating the problem:** We are given the equations $x=\frac{4}{y}$ and $\frac{y}{z}=8$, and we need to find the value of $\frac{z}{y}$. 2. **Understanding the given equations:** - From $x=\frac{4}{y}$, we know $x$ is related to $y$, but this is not directly needed to find $\frac{z}{y}$. - From $\frac{y}{z}=8$, we have a direct relationship between $y$ and $z$. 3. **Using the second equation:** $$\frac{y}{z} = 8$$ This means $y = 8z$. 4. **Finding $\frac{z}{y}$:** We want to find $\frac{z}{y}$. Using $y=8z$, substitute into $\frac{z}{y}$: $$\frac{z}{y} = \frac{z}{8z}$$ 5. **Simplify the fraction:** Since $z \neq 0$, we can cancel $z$ in numerator and denominator: $$\frac{z}{8z} = \frac{1}{8}$$ **Final answer:** $$\boxed{\frac{z}{y} = \frac{1}{8}}$$